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ARITHMETIC THE SECOND
(B) There may be a number in the upper class which is less
than every other number in that class.
(C) Xeither of the numbers (greatest in [A], least in [B])
described in (A), (B) may exist.
The possibility which leads to irrational numbers is (C), For
if (C) holds, the assumed rule 'defines' a definite break or 'cut*
in the set of all rational numbers. The upper and lower classes
strive, as it were, to meet. But in order for the classes to meet
the cut must be filled with some 'number', and, by (C), no such
filling is possible.
Here we appeal to intuition. All the distances measured from
any fixed point along a given straight line 'correspond' to
^numbers' which 'measure' the distances. If the cut is to be left
unfilled, we must picture the straight line, which we may con-
ceive of as having been traced out by the continuous motion of
a point, as now having an unbridgeable gap in it. This violates
our intuitive notions, so we say, by definition, that each cut
does define a number. The number thus defined is not rational,
namely it is irrational. To provide a manageable scheme for
operating with the irrationals thus defined by cuts (of the kind
[C]) we now consider the lower class of rationals in (C) as being
equivalent to the irrational which the cut defines.
One example will suffice. The irrational square root of 2 is
defined by the cut whose upper class contains all the positive
rational numbers whose squares are greater than 2, and whose
lower class contains all other rational numbers.
If the somewhat elusive concept of cuts is distasteful two
remedies may be suggested: devise a definition of irrationals
which is less mystical than DedeMnd's and fully as usable;
follow Kronecker and, denying that irrational numbers exist*
reconstruct mathematics without them. In the present state of
mathematics some theory of irrationals is convenient. But,
from the very nature of an irrational number, it would seem to
be necessary to understand the mathematical infinite
thoroughly before an adequate theory of irrationals is
possible. The appeal to infinite classes is obvious in Dede-
kind's definition of a cut. Such classes lead to serious logical